FIG. 1 shows a conventional method for separating a multi-component signal. A mixture of amplitude-frequency modulated signals (AM-FM) and sinusoidal signals occurs frequently in acoustics applications, biological systems, and as signals received by vehicular collision avoidance radars that transmit continuous wave frequency modulation (CWFM) signals.
A single component AM-FM sinusoidal signal is represented asx(t)=A(t)cos(2πft+φ(t)),  (1)where A(t) indicates a time-varying amplitude envelope and φ is the phase angle. A K multi-component signal 110 is given byy(t)=Σi=1KAi(t)cos(2πfit+φi(t)),0<t<T  (2)where T is a signal duration.
Gianfelici et. al., in “Multicomponent AM-FM Representations: An Asymptotically Exact Approach,” IEEE Trans. Audio, Speech and Language Processing, vol. 15, no. 3, March 2007, describe a method called an Iterated Hilbert Transform (IHT). Generally, the Hilbert transform is a linear operator that takes a function, u(t), and produces a function, H(u)(t) in the same domain. The IHT can be used to estimate instantaneous frequencies of the components 150 of the signal in equation (2). The performance of IHT is suboptimal when the amplitude of a component is within a close range, e.g., A2/A1=2, in a two component case. The IHT is followed by a Teager-Kaiser energy detector (TKED) based frequency estimator 160, which outputs 170 direct current (DC) component signals.
Santhanam et al., in “Multicomponent AM-FM Demodulation via Periodicity-based Algebraic Separation and Energy-based Demodulation,” IEEE Trans. Commun., vol. 48. no. 3, March 2000, describe a method called PASED 150, which is a non-linear method that can separate mixed periodic signals with similar strengths.
PASED works well even when the signals have a small spectral separation. However, PASED needs to know both the period of each signal component and the number of components in the mixture.
Therefore, PASED is generally prefaced by a Double Differencing Function (DDF) 120 to estimate the parameters of the input signal. The parameters include the number of components and their periods. With noisy signals, DDE is also suboptimal.
Therefore, the conventional PASED 130 may not output an optimal separation in low signal-to-noise-ratio (SNR) cases. The PASED also uses a zero DC constraint 140 for each sinusoid expected to be in the multi-component signal. If the signal is not integrated over the correct period of a sinusoid, then the DC-constraint is violated, because the integration does not result in a zero value.